Geometric moduli theory and its theoretical applications

In this project we aimed at studying global structures of certain geometric spaces so that we may apply them to the related mathematical theories. The main results of our studies are 1) construction of the second compactifications of moduli spaces of abelian varieties, and study of the relation with the other important compactifications, 2) proof of Riemann hypothesis for some of zeta functions of the moduli spaces of semi-stable vector bundles over an algebraic curve, 3) a characterization of one of Painleve differential equations through the study of stable vector bundles of rank two, 4) proof of the isomorphism between the quantum cohomology ring and the Jacobi ring of a potential in mirror symmetry through the study of the moduli space of Lagrangian submanifolds of a toric manifold, 5) generalization and further study of Arrow's impossibility theorem in statistical economics in terms of hyperplane arrangement.